Hall Ticket No Question Paper Code:AHSC07
INSTITUTE OF AERONAUTICAL ENGINEERING
(Autonomous)
Dundigal-500043, Hyderabad
B.Tech II SEMESTER END EXAMINATIONS (REGULAR) - SEPTEMBER 2022 Regulation:UG20
MATHEMATICAL TRANSFORM TECHNIQUES
Time: 3 Hours (Common to AE | ECE | EEE |ME | CE ) Max Marks: 70 Answer ALL questions in Module I and II
Answer ONE out of two questions in Modules III, IV and V All Questions Carry Equal Marks
All parts of the question must be answered in one place only
MODULE – I 1. (a) Using Laplace transform, solve d2y
dt2 +dy
dt =t2+ 2tgiven that y(0)=4, y0(0) =−2.
[BL: Apply| CO: 1|Marks: 7]
(b) Using convolution theorem find the inverse Laplace transform of 4 (s2+ 2s+ 5)2
[BL: Apply| CO: 2|Marks: 7]
MODULE – II
2. (a) If F(S) and G(S) are the Fourier transforms of f(x) and g(x) respectively then prove the following i)F(af(x) +bg(x)) =aF(S) +bG(S)
ii) Fs(f(x)cosax) = 1/2(Fs(S+a) +Fs(S−a)) [BL: Apply| CO: 3|Marks: 7]
(b) Express the functionf(x) =
(1 |x| ≤1
0 x >1 as a Fourier integral. Hence, evaluate i)
R
∞0
sinλcosλx
λ dλ
ii)
R
∞ 0sinλ
λ dλ [BL: Apply| CO: 3|Marks: 7]
MODULE – III
3. (a) Using double integrals show that the area between the parabolasy2= 4axand x2 = 4ayis 16 3 a2 [BL: Apply| CO: 4|Marks: 7]
(b) Evaluate
R
∞ 0R
∞0 e−(x2+y2)dxdyby changing to polar coordinates and hence evaluate
R
∞0 e−x2dx.
[BL: Apply| CO: 4|Marks: 7]
4. (a) Solve
R
4 0R
2√ z 0R
√4z−x20 dxdydz [BL: Apply| CO: 4|Marks: 7]
(b) Evaluate
R
C
−
→A .−→ dr if−→
A = (3x2+ 6y)−→
i −14yz−→
j + 20xz2−→
k and C is the curvex=t, y=t2, z=t3
from (0,0,0)to(1,1,1). [BL: Apply| CO: 4|Marks: 7]
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MODULE – IV 5. (a) Find the value of ‘a’ if the vector field (ax2y+yz)−→
i + (xy2−xz2)−→
j + (2xy−2x2y2)−→
k has zero divergence. Find the curl of the vector field when it has zero divergence.
[BL: Apply| CO: 5|Marks: 7]
(b) Evaluate by Green’s theorem
R
Ce−x(sinydx +cosydy) where C is the rectangle with vertices (0,0),(π,0),(π,π
2),(0,π
2) [BL: Apply| CO: 5|Marks: 7]
6. (a) Using the Gauss divergence theorem, evaluate
R R
S
−
→F .−→n dS, where −→
F =x3−→
i +y3−→
j +z3−→ k and S is the spherex2+y2+z2=a2. [BL: Apply| CO: 5|Marks: 7]
(b) Evaluate
R
C(xydx+xy2dy) by Stoke’s theorem, C is the square in the XY-plane with vertices
(1,0),(−1,0),(0,1),(0,−1). [BL: Apply| CO: 5|Marks: 7]
MODULE – V
7. (a) Find the singular solution ofz=px+qy. [BL: Apply| CO: 6|Marks: 7]
(b) Solve x2(y−z)p+y2(z−x)q =z2(x−y) [BL: Apply| CO: 6|Marks: 7]
8. (a) Form partial differential equation by eliminating the arbitrary constant a and b from the equations i)2z= x2
a2 + y2 b2
ii) z= (x−a)2+ (y−b)2+ 1 [BL: Apply| CO: 6|Marks: 7]
(b) Find the partial differential equation px+qy=pq by Charpit method.
[BL: Apply| CO: 6|Marks: 7]
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